# IGCSE Higher: Circles and Sectors

Scheme of work: IGCSE Higher: Year 10: Term 4: Circles and Sectors

#### Prerequisite Knowledge

1. Finding the Area of a Circle
• Concept: Knowing how to calculate the area of a circle using the formula.
• Example: $\text{Area} = \pi r^2 \quad \text{where } r \text{ is the radius of the circle}$
2. Finding the Circumference of a Circle
• Concept: Knowing how to calculate the circumference of a circle using the formula.
• Example: $\text{Circumference} = 2\pi r \quad \text{where } r \text{ is the radius of the circle}$

#### Success Criteria

1. Finding Perimeter and Area of Sectors
• Performing the Calculation: Students should be able to calculate the perimeter and area of sectors of circles.
• Example: $\text{Area of a sector} = \frac{\theta}{360} \times \pi r^2, \quad \text{Perimeter of a sector} = 2r + \frac{\theta}{360} \times 2\pi r$
• Describing the Calculation: Students should be able to describe how to use the sector formulas to find the perimeter and area.
• Example: $\text{To find the area of a sector, use } \frac{\theta}{360} \times \pi r^2 \text{ where } \theta \text{ is the angle of the sector and } r \text{ is the radius}$
2. Manipulating the Sector Formulae
• Performing the Manipulation: Students should be able to manipulate the sector formulae to find a different variable when the area or circumference is known.
• Example: $\text{If the area of a sector is known, find the radius using } r = \sqrt{\frac{\text{Area} \times 360}{\theta \times \pi}}$
• Describing the Manipulation: Students should be able to describe how to rearrange the sector formulas to solve for a different variable.
• Example: $\text{To find the radius when the area is known, rearrange the formula to } r = \sqrt{\frac{\text{Area} \times 360}{\theta \times \pi}}$
3. Problem Solving with Area and Perimeter
• Applying the Formulas: Students should be able to apply the formulas to solve problems involving the area and perimeter of circular and compound shapes involving circles.
• Example: $\text{Find the area of a compound shape consisting of a rectangle and a semicircle}$
• Describing the Solution: Students should be able to describe how to combine the areas of different shapes to find the total area.
• Example: $\text{To find the total area, sum the area of the rectangle and the area of the semicircle}$

#### Key Concepts

1. Understanding Sector Area
• Concept: Knowing that the area of a sector is a fraction of the area of the whole circle.
• Example: $\text{Area of a sector} = \frac{\theta}{360} \times \pi r^2$
• Describing Sector Area: Understanding that the fraction is determined by the central angle of the sector.
• Example: $\theta \text{ is the central angle in degrees, and } r \text{ is the radius}$
2. Understanding Sector Perimeter
• Concept: Knowing that the perimeter of a sector includes the arc length and the two radii.
• Example: $\text{Perimeter of a sector} = 2r + \frac{\theta}{360} \times 2\pi r$
• Describing Sector Perimeter: Understanding that the arc length is a fraction of the circumference of the whole circle.
• Example: $\theta \text{ is the central angle in degrees, and } r \text{ is the radius}$
3. Manipulating Formulas
• Concept: Knowing how to rearrange formulas to solve for a different variable.
• Example: $r = \sqrt{\frac{\text{Area} \times 360}{\theta \times \pi}}$
• Describing Manipulations: Understanding the steps to isolate the variable of interest.
• Example: $\text{To find the radius, rearrange the formula for the area of a sector}$
4. Problem Solving
• Concept: Applying area and perimeter formulas to solve real-world problems.
• Example: $\text{Calculate the area of a garden with circular sections}$
• Describing Solutions: Understanding how to decompose complex shapes into simpler parts.
• Example: $\text{Sum the areas of individual sections to find the total area}$

#### Common Misconceptions

1. Calculating Sector Area
• Common Mistake: Students might forget to convert the angle to degrees or use the wrong fraction.
• Example: Calculating the area of a sector with a 60-degree angle as: $\text{Incorrect: } \frac{60}{180} \times \pi r^2$ $\text{Correct: } \frac{60}{360} \times \pi r^2$
2. Calculating Sector Perimeter
• Common Mistake: Students might forget to include the two radii in the perimeter calculation.
• Example: Calculating the perimeter of a sector with a 60-degree angle as: $\text{Incorrect: } \frac{60}{360} \times 2\pi r$ $\text{Correct: } 2r + \frac{60}{360} \times 2\pi r$
3. Manipulating Formulas
• Common Mistake: Students might incorrectly rearrange the formula, leading to incorrect solutions.
• Example: Rearranging the formula for the area of a sector incorrectly to find the radius: $\text{Incorrect: } r = \sqrt{\frac{\text{Area} \times \theta}{360 \times \pi}}$ $\text{Correct: } r = \sqrt{\frac{\text{Area} \times 360}{\theta \times \pi}}$
4. Problem Solving
• Common Mistake: Students might incorrectly combine areas or perimeters of compound shapes.
• Example: Miscalculating the area of a compound shape consisting of a rectangle and a semicircle: $\text{Incorrect: } \text{Area} = \text{Area of rectangle} + \text{Circumference of semicircle}$ $\text{Correct: } \text{Area} = \text{Area of rectangle} + \frac{1}{2} \times \pi r^2$
• Common Mistake: Students might forget to convert units when necessary.
• Example: Calculating the area in square meters when the radius is given in centimeters without converting the units.

### Mr Mathematics Blog

#### Estimating Solutions by Rounding to a Significant Figure

Explore key concepts, FAQs, and applications of estimating solutions for Key Stage 3, GCSE and IGCSE mathematics.

#### Understanding Equivalent Fractions

Explore key concepts, FAQs, and applications of equivalent fractions in Key Stage 3 mathematics.

#### Transforming Graphs Using Function Notation

Guide for teaching how to transform graphs using function notation for A-Level mathematics.